Vol 16 No 11, November 2007 1009-1963/2007/16(11)/3256-06

Chinese Physics

c 2007 Chin. Phys. Soc.

and IOP Publishing Ltd

Notch filter feedback controlled chaos in buck converter∗ Lu Wei-Guo(©•I)† , Zhou Luo-Wei(±X‘), and Luo Quan-Ming(Û²) Key Laboratory of High Voltage Engineering and Electrical New Technology of Ministry of Education, College of Electrical Engineering, Chongqing University, Chongqing 400044, China (Received 18 March 2007; revised manuscript received 9 April 2007) A method of controlling chaos in the voltage-mode buck converter is presented by using an improved notch filter feedback control in this paper. The proposed control part comprises a notch filter and a low-pass filter. The discrepancy between the outputs of the two filters is introduced into the control prototype of the power converter. In this way, the system period-1 solution is kept unchanged. The harmonic balance method is applied to analysing the variation law of the system bifurcation point, and then the stable range of the feedback gain is ascertained. The results of simulation and experiment are also given finally.

Keywords: chaos, control of chaos, buck converter, notch filter PACC: 0545

1. Introduction Bifurcation and chaos commonly exist in DC–DC converters with Feedback control,[1−6] which deteriorate the performance of the power converters, for instance, output ripple, conversion efficiency.[7] Furthermore there are few effective approaches to precisely forecasting and controlling the state variable or the output in chaos. Therefore the effective way of suppressing chaotic behaviour in DC–DC converter is of great significance in engineering. The voltage-mode buck converter is the earliest power converter for the investigation of chaotic behaviour. The output voltage changes into a chaotic state from stable period-1 with the variation of its input voltage. The output behaves in a quasirandom way, which deteriorates the system performance. Some papers devoted to the controlling chaos in this kind of system have been published. In Refs.[7–9], chaotic state in a DC–DC converter was controlled into period-1 via an external periodic signal perturbing the system parameters. Thus the perturbing signal circuits are necessary, for the quantitative analysis of the control parameters is quite difficult. Another way to control chaos is to adopt the output or state variable feedback. In Ref.[10], time-delay feedback was first applied to the control of chaos in the voltage-mode buck converter. The control parameters were ascertained ∗ Project

analytically by using the Floquet theory. However, the control parameter is not only an analytic solution from the viewpoint of stability. In Refs.[11,12], the voltage derivative feedback and state variable feedback were applied to the control of chaos in the voltage-mode buck converter. Both of them were hardly realized in engineering because the feedback signals are introduced into the power circuits. Another advanced chaos control method is based on filters. In Ref.[13], the control of chaos in the buck converter was achieved by using a washout filter which is a kind of high-pass filter. The band-stop notch filter can also effectively control the chaotic system,[14] but its application to a switching power converter has not been reported to date. From the above analyses, this paper aims to apply the output notch filter feedback to the control of chaos in the voltage-mode buck converter. The notch filter frequency is set to be the switching frequency of the power converter because the period-1 is our control target. Thus, only the feedback gain needs to be ascertained. The stable range of the feedback gain is obtained by applying the harmonic balance method to the analysis of the bifurcation point variation law. In this way, the system stable range can be widened, and the output in steady state or period-1 is kept unchanged.

supported by the National Natural Science Foundation of China (Grant No 50677071). [email protected] http://www.iop.org/journals/cp http://cp.iphy.ac.cn

† E-mail:

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Notch filter feedback controlled chaos in buck converter

2. Notch filter feedback controlled buck converter In the voltage-mode buck converter, the variation of its input voltage can cause the occurrence of perioddoubling bifurcation, and then the output changes into a chaotic state.[6,10,11] Here the sub-harmonic, whose frequency is less than the switching frequency, will exist in this system. Notch filter can suppress the occurrence of sub-harmonic.[13] Therefore it can stabilize the system in period-1 by adding a notch filter to the feedback control of the buck converter. The corresponding transfer function of the feedback notch filter is s2 + ωn2 Gf (s) = p 2 , (1) s + 2ξωn s + ωn2 where ξ is the damping factor, p is the feedback gain, and ωn is the notch frequency. As we expect the system to be controlled in period-1, ωn should be set to be the switching frequency of the power converter. While the system is in period-1, s is equal to j2π/T , where T is the switching period of the power converter. So here the relation s2 + ωn2 = 0 is satisfied, and we have Gf (j2πT ) = 0.

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Therefore, the notch filter output in period-1 is a constant steady-state component. The control signal in a steady-state should approach zero so as to keep the period-1 solution of the system to be controlled unchanged. This expectation can promise well, if we choose the feedback control signal as the difference signal between the outputs of the notch filter and a low-pass filter. And here the low-pass filter is used to filter out the average steady-state component from the output voltage. The schematic diagram and the block diagram of the voltage-mode buck converter with notch filter (NF) feedback control are shown in Figs.1(a) and 1(b) respectively, where the additional control is framed by the dot line, which comprises a notch filter and a lowpass filter (LPF), with Vin being the input voltage, Vref the reference voltage, vramp the saw-tooth voltage, vd the diode voltage, vcon the original feedback control ′ voltage, vcon the new feedback control voltage, ∆vcon the output voltage of additional control part, and k1 and k2 , respectively, the feedback gains for the original control and for notch filter control. When ∆vcon is equal to zero, the system turns into a conventional voltage-mode buck converter.

Fig.1. The voltage-mode buck converter with notch filter feedback control. Subfigure (a) shows the schematic diagram and (b) the block diagram.

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′ vcon (T + dT − δ) = vramp (T + dT − δ).

From Figs.1(a) and 1(b), we have vcon = k(Vref − vo ),

(2)

′ vcon = vcon − ∆vcon .

(3)

(10)

We define the frequency domain variable corresponding to vo (t) as vo (s) and do the same to the other time domain variables. G1 (s) is the transfer function of the filter which is composed of L, R and C as shown in Fig.1(a), so we have G1 (s) =

vo (s) 1 = . 2 vd (s) LCs + (L/R)s + 1

(4)

The transfer function of the notch filter feedback control is G2 (s) as shown in Fig.1(b), which can be obtained as fellows: ∆vcon (s) = k2 vo (s)(Gf (s) − G3 (s)) = k2 G1 (s)Vd (s)(Gf (s) − G3 (s)), G2 (s) =

(5)

∆vcon (s) = k2 G1 (s)(Gf (s) − G3 (s)), (6) Vd (s)

where G3 (s) is the transfer function of the low-pass filter, which is written as 2π/(k3 T ) . G3 (s) = s + 2π/(k3 T )

Fig.2. The waveforms of the feed voltage, saw-tooth voltage and pulse drive signal in period-1 (a) and period-2 (b).

Fourier series decomposition is applied to vd (t) in either in period-1 or period-2, and we have in period-1 vd (t) =

(7)

As is known, the larger the value of k3 , the better the performance of the low-pass fitter, but the worse the dynamic property of the system. So the value of k3 should be appropriately chosen. And we define the hybrid transfer function as

+∞ X

cn1 e j n1 ωs t ,

(11)

n1 =−∞

where

Vin (1 − e −j n1 ωs dT ), j2n1 π ωs = 2π/T , in period-2 cn1 =

vd (t) =

+∞ X

cn2 e (1/2)j n2 ωs t ,

(12)

n2 =−∞

G(s) = k1 G1 (s) − G2 (s).

(8)

In the following, we will apply the harmonic balance method[15] to analysing the bifurcation point variation law in fore and aft the notch filter feedback control. Figures 2(a) and 2(b) show the wave′ forms of the feed voltage vcon , saw-tooth voltage vramp and pulse drive signal in period-1 and period-2 respectively, where vramp = Vl + (Vh − Vl ) mod (t/T , 1). From Fig.2, the following relationship can be obtained, wherein d is the duty ratio of the power converter: in period-1 ′ vcon (dT )

where while n2 is odd number, cn2 =

Vin (e −(1/2)jn2 ωs (dT −δ) − e −(1/2)jn2 ωs (dT +δ) ), j2n2 π

while n2 is even number cn2 =

Vin (2 − e −(1/2)jn2 ωs (dT +δ) j2n2 π − e −(1/2)jn2 ωs (dT −δ) ).

Thus, in periodic-1 there exists ′ vcon (dT ) = k1 Vref −

+∞ X

n1 =−∞

= k1 Vref − c10 − = vramp (dT ),

in period-2

cn1 e jn1 ωs dT G(jn1 ωs )

(9) × Im

V  in

π

+∞ X  (e j n1 ωs dT − 1) G(jn1 ωs ) , (13) n1 n =1 1

′ vcon (dT

+ δ) = vramp (dT + δ),

where c10 = [vd (t)]ave = dVin G(0).

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Notch filter feedback controlled chaos in buck converter

Combining expression (9) with expression (13), the relationship between Vin and d can be obtained as follows: k1 Vref −vramp (dT ) Vin = . (14) P (e j n1 ωs dT−1) 1 G(jn ω )] dG(0)+( π )Im[ +∞ 1 s n1 =1 n1 In a similar way, combining expression (10) with

N=

 1 1  G j n2 − ωs sin((2n2 − 1)ωs δ), 2n2 − 1 2

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expression (12), the relationship between Vin and d in period-2 can be obtained as follows: Vin =

πδ(Vh − Vl )/T , P −Re[ +∞ n2 =1 (N + M )]

(15)

where

and M =

1 G(jn2 ωs )(2e j n2 ωs d sin(n2 ωs δ) − sin(2n2 ωs δ)). 2n2

When δ in expression (15) approaches zero, the input voltage bifurcation point (the critical value between period-1 and period-2) is Vin∗ =

(Vh − Vl ) . P+∞ −2Re[ n2 =1 (G(j(n2 − 21 )ωs ) + G(jn2 ωs )(e j n2 ωs dT − 1)]

Considering G(s) as a low-pass filter, the denominator term to the right of expression (16) can be approximated as a first-order term, and then we have Vin∗ ∝

Vh − Vl . 2Re[(G(jωs /2)]

(16)

Based on the above analyses, the notch filter feedback control is considered not to change the steadystate solution and the period-1 solution, which is vital to the control of chaos in power converters.

(17)

The conclusions can been drawn from expression (17). As vramp changes, the bifurcation point will change. The change of the parameters in G(s) will also affect the bifurcation point. So the notch filter feedback control, corresponding to G2 (s) included in G(s), can also affect the bifurcation point. The bifurcation point is enlarged, and the chaotic behaviour is suppressed by choosing an appropriate value of k2 . Combining expression (14) with expression (16), the corresponding bifurcation points Vinb for different values of k2 can be obtained by numerical computation. But it is necessary to choose values large enough for the Fourier series numbers n1 and n2 . The value of k2 , which ensures the system in period-1, is ascertained according to the bifurcation point variation law. The influence on the output voltage in steadystate is discussed as below. When an additional control stabilizes the system in period-1, we have G2 (0) = k2 (Gf (0) − G3 (0)) = 0. And when k3 is large enough, we have |G3 (j2π/T )| = 1/k3 ≈ 0, Gf (j2π/T ) = 0, |G2 (j2π/T )| = |k2 (Gf (j2π/T ) − G3 (j2π/T ))| ≈ 0.

3. Simulation results In this section, the simulation work will be carried out according to the buck converter schematic diagram shown in Fig.1(a), and the parameters in simulation are in agreement with those in Ref.[3], specifically chosen as follows: Vin = 20V − 35 V, L = 20 mH, R = 22Ω, C = 47 µF, vramp = 3.8 + (8.2 − 3.8)mod (t/T, 1) V, Vref = 11.3 V, k1 = 8.4, and T = 0.4 ms. Furthermore, the parameters of the notch filter and the low-pass filter are chosen as follows: ξ = 0.707, p = 1, ωn = 2π/T , and q = 2π/(25 T ). Figure 3(a) shows the output bifurcation diagram with an input voltage used as a bifurcation parameter while the notch filter control is invalid, i.e. k2 is equal to zero. Figure 3(b) shows the Lyapunov exponential spectrum for Vin = 24V − 34 V and k2 = 0. It is observed that the system changes into period-2 while the input voltage Vin is about 24.5 V. And the system is in chaos and the corresponding Lyapunov exponent (LE) is greater than zero as shown in Fig.3(b), while Vin is greater than 31.5 V. The input voltage bifurcation point value should be greater than 35 V to ensure the system in period-1 for Vin = 20V−35 V. Thus, the range of k2 is ascertained according to the expectant bifurcation point.

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Fig.3. The output bifurcation diagram (a) and the Lyapunov exponent (LE) for k2 = 0 (b).

According to Section 2, combining expression (14) with expression (16), we set n1 and n2 to be twenty and use numerical method to obtain the bifurcation point value. Figure 4 shows the bifurcation point diagram for k2 = 0 − 5. The bifurcation point value is about 24.5 V for k2 = 0, which is in agreement with the result in the bifurcation diagram shown in Fig.3(a). From Fig.4, we know that it can stabilize the system in period-1 while Vin is equal to 34 V and k2 is greater than 3.5. Figures 5(a)–5(c) show the capacitive voltage, the inductive current and the feed control voltage of the buck converter while the additional control takes effect at t = 0.03 s for Vin = 34 V and k2 = 5.

Fig.4. The bifurcation point value Vin versus feedback gain k2 .

It can be seen from Fig.5 that the proposed method can change the chaotic state of the system to be controlled into period-1 quickly. ∆vcon in steadystate is closer to zero, so the period-1 solution is kept unchanged.

Fig.5. The simulating waveforms for k2 = 5 of the capacitive voltage (a), the inductive current (b), and the feedback control voltage (c).

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Notch filter feedback controlled chaos in buck converter

4. Experimental results The experimental circuits are built up according to the schematic diagram shown in Fig.1(a), where the notch filter circuit is given by the Wien-bridge and the low-pass filter circuit is constructed by a resistance and a capacitance in series. The experimental parameters are in agreement with those in simulation. Figure 6(a) shows the experimental waveforms of the capacitive voltage (CH1:2V/div) and the induc-

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tive current (CH2:500mA/div), and Figure 6(b) shows the experimental waveforms of the original feedback control voltage (CH1:10V/div) and the output of the notch filter feedback control (CH2:10V/div). Without the notch filter feedback control, the voltage and the current are in chaos. On the contrary, the system changes into period-1 from the chaotic state, the voltage and current ripple decrease and the controlled period-1 behaviour acts as that in unstable periodic orbit (UPO) of the original system.

Fig.6. The experimental waveforms for Vin = 34 V and k2 = 5 of the inductive current and the capacitive voltage (a), and the feedback control voltage and the output of the notch feedback control (b).

5. Conclusion In this paper we apply the notch filter feedback to the control of chaos in a voltage-mode buck converter, and the control effect is equivalent to that of the time-delay feedback control. It can not only con-

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trol the chaotic buck converter into period-1 quickly, but also keep the period-1 solution unchanged. And either the notch filter or the low-pass filter adopted in this method is realized by analogue circuits easily. Meanwhile, this method can be deployed to control the chaos in other power converters.

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